The theory of the firm and industry equilibrium

Martin J. Osborne
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6.2 Price-discriminating monopolist

Suppose that each buyer can purchase either 0 or 1 unit of the good, and that buyers differ in their reservation values. In this case a monopolist may be able to do better than it can by setting a single price: she may be able to sell at high prices to buyers with high reservation prices and to low prices to buyers with low reservation prices.

Perfect price discrimination

If the monopolist can identify buyers by their reservation values and set different prices for different buyers, and buyers do not have the possibility of trading between themselves, then even if it sets a price for each buyer just below her reservation value, that buyer will still purchase the good. Assuming that a buyer who is faced with a price equal to her reservation value buys the good (she is indifferent between doing so and not buying) then the monopolist's optimal strategy is clear: set a price for each buyer equal to her reservation value.

(Note that if the buyers can trade between themselves then this strategy may come unstuck: buyers with low reservation values may be able to buy many units of the good, and re-sell some of them to buyers with high reservation values. In some cases, a monopolist can prevent such trades. For example, in the case of admission to a movie, admission may be granted to an adult only if she is in possession of an adult ticket.)

How much should the monopolist produce in this case? So long as there is a buyer whose reservation price exceeds the monopolist's marginal cost, it is in the interest of the monopolist to sell to that buyer. Thus the monopolist will produce the output y for which MC(y) is equal to P(y). That is,

the optimal output of a perfectly discriminating monopolist is Pareto stable!
In this outcome the monopolist gets all the surplus, so unless the monopolist is needy the outcome is not likely to be equitable---but it is Pareto stable.

Ordinary price discrimination

A monopolist cannot usually discriminate perfectly between buyers, since it does not know each buyer's reservation price. Nevertheless, it may be able to charge different prices to different types of buyer, achieving some degree of discrimination. (For example, Bell Canada charges different prices to businesses and to individuals; airline charge different prices to people who can reserve in advance and those who cannot.)

Suppose that the monopolist can separate the market into two parts, one in which the inverse demand function is P1 and one in which it is P2. Then its total revenue functions in the two markets are TR1(y) = yP1(y) and TR2(y) = yP2(y). The monopolist's profit-maximization problem in this case is to choose outputs y1 and y2 for the markets to solve

maxy1,y2 π(y1,y2) = maxy1,y2 [TR1(y1) + TR2(y2) − TC(y1 + y2)].

At a solution to this problem the value of y1 must maximize profit, given the value of y2, and the value of y2 must maximize profit, given the value of y1. Thus the following conditions must be satisfied:

MR1(y*1) − MC(y*1 + y*2) = 0

MR2(y*2) − MC(y*1 + y*2) = 0.
We can write these conditions alternatively as
MR1(y*1) = MR2(y*2) = MC(y*1 + y*2).
(As in the case of a monopolist setting a single price, values of y*1 and y*2 that satisfy these equations may correspond to minimum profit rather than maximum profit, and if the monopolist's profit is negative when it chooses y*1 and y*2 then it should instead produce nothing.)

Relation with the elasticity of demand

We have MR(y) = P(y)[1 − 1/|η(y)|], where η(y) is the elasticity of demand at y. Thus if y*1 and y*2 satisfy the conditions above then
p1[1 − 1/|η1(y*1)|] = p2[1 − 1/|η2(y*2)|].
Suppose that the elasticities are constant, independent of demand. Then the condition is
p1[1 − 1/|η1|] = p2[1 − 1/|η2|],
or
p1
p2
 =
1 − 1/|η2|
1 − 1/|η1|
.

Thus if |η1| > |η2| then p1 < p2. [Try η1 = −4 and η2 = −2: then p1/p2 = (1−1/2)/(1 − 1/4) = (1/2)(4/3) = 2/3.]

That is:

a monopolist charges a lower price in a market in which demand is more elastic.
Example: ordinary price discrimination
In market 1 we have Qd(p) = 10 − p/2 while in market 2 we have Qd(p) = 32 − 2p. The monopolist's total cost function is TC(y) = y2. What outputs does the monopolist sell in each market?

We have

TR1(y1) = y1(20 − 2y1)
TR2(y2) = y2(16 − y2/2)
and hence
MR1(y1) = 20 − 4y1
MR2(y2) = 16 − y2.

Thus the condition MR1(y*1) = MR2(y*2) = MC(y*1 + y*2) is equivalent to

20 − 4y*1 = 16 − y*2 = 2(y*1 + y*2).
Isolating y*2 in the first equation we obtain
y*2 = 4y*1 − 4.
Plugging this into the expression 20 − 4y*1 = 2(y*1 + y*2) we obtain
20 − 4y*1 = 2(y*1 + 4y*1 − 4)
or
28 = 14y*1
or
y*1 = 2,
so that
y*2 = 4.
(I am assuming that the first-order conditions identify a maximum rather than a minimum.)